Ab Initio Molecular Dynamics with Born-Oppenheimer and Extended Lagrangian Methods Using Atom Centered Basis Functions
H. Bernhard Schlegel
Departmentof Chemistry, Wayne State University, Detroit,Michigan48202, USA ReceivedFebruary 25, 2003
In ab initio molecular dynamics, whenever information about the potential energy surface is needed for integratingthe equations of motion, itis computed “onthefly” using electronic structure calculations. For Born-Oppenheimer methods, the electronic structure calculations are converged, whereas in the extended Lagrangianapproachtheelectronic structureis propagated along with the nuclei. Some recent advances for both approaches arediscussed.
Key Words : Ab initio molecular dynamics, Direct classical trajectory calculations, Born-Oppenheimer dynamics,Extended Lagrangiandynamics,ADMP
Introduction
As discussed in numerous chapters andmonographs,1-16 classicaltrajectoriesof moleculesmovingon potential energy surfacesprovide a wealth of informationabout reactivityand dynamics. Because molecular dynamics calculations may involve extensive sampling of initialconditions and / or long simulation times, the molecular energy and its derivatives need tobecomputedfrequently during the integration of the equations of motion. Traditionally, such studies have used analyticpotential energy surfaces fittedtoexperimental and computational data.Potentialenergy surfaces obtainedfrom well parameterized molecular mechanicscalculationscan be quite satisfactory for simulationsnearequilibrium.However, for reactivesystems, specific potential energy surfaces must be devised for each unique system. Constructing potential energy surfaces by fittingto experimentaldata and / orab initio molecularorbitalenergiescan beboth tedious and full of pitfalls.17,18 Alternatively, ab initio or semi-empirical molecular orbital calculations can beuseddirectly to obtain the energies and derivatives as theyare needed, thus avoiding thefittingprocess.19 Thisapproach hasbeen termedab initio moleculardynamics(AIMD). The calculationoftrajectories by AIMD methods is a comparatively new area19 and is expanding rapidly as affordability of computer power increases and moreefficientsoftware is developed.
Directclassical trajectorycalculationscan begrouped into two major categories: Born-Oppenheimer(BO)methods and extended Lagrangian (EL) methods. For the former, the electronic structure calculation is fully converged in the Born-Oppenheimer (clamped nuclei) approximation, each time that information aboutthe potential energy surface is neededfor a given nuclearconfiguration. In the extended Lagrangian approach, both the wavefunction and thenuclei are treated as dynamic variables. With an appropriate adjustment of the time scales for the dynamics of the wavefunction, both can be propagated satisfactorily with Lagrangian equations of motion, withoutthe extrawork of converging the wavefunction at each step. The resulting
dynamics ofthenucleiare comparable to thatobtainedwith theBorn-Oppenheimer approximationbut at lower cost.The Car-Parrinello method is the archetypical example of this approach.20 The present overview is not intended to be a thorough review of the field, but is concerned only with some highlights of recentcontributions to the development ofAIMDmethodsfrom achemical perspective.
Results andDiscussion
Born-Oppenheimer methods.Thesimplest approach for Born-Oppenheimer dynamics useselectronic structuremethods to calculatetheenergy and gradients directly. Methods such as velocity Verlet, fourth order Runge-Kutta, sixth order Adams-Moulton-Bashforth and related predictor-corrector algorithms21 are typical gradient-based methods used to integrate the equations of motion. Because this class of integratorsrequiresfairly small time stepsto determine the trajectories accurately, many thousands ofelectronic structure calculations may be needed, even for fairly fast reactions.
Code for calculating classicaltrajectories has been incorpo
rated into anumber ofwidelydistributedelectronic structure packages (Dalton, DMol, Gamess, Gaussian, HyperChem, NWChem,etc.). Alternatively, a standardelectronic structure package can be called as a subroutine from a classical trajectory code.
Analytic secondderivatives ofthe energy (Hessians) can be calculated readily for a number of electronic structure methods, includingHartree-Fock (HF), multi-configuration SCF(MCSCF),densityfunctional theory (DFT) and second orderMoller-Plessetperturbationtheory (MP2). Thegradient and Hessian provide a local quadratic approximation to the potentialenergy surface and the equations of motion canbe integrated on this local surface in closed form, allowing significantlylarger steps between electronic structure calcu
lations than for gradient-basedmethods. This approachwas pioneered by Helgaker, Uggerud and Jensenin their studies of H2 + H and CH2OH tHCO+ + H2 atthe MCSCF levelof theory.22,23 Numerous systems have now been studied by
Calculate the energy,
Figure 1. Hessian-based predictor-corrector algorithm for integrat
ing trajectories on the Born-Oppenheimer surface.
these authors with the secondorderHessian-based trajectory integration method, includingC2H6+,H3O+ + NH3, CH2NH2+, NHNH2+, NH2NH3+ and HNO +HNO.24-30
Figure 1 illustrates a Hessian-based predictor-corrector method that we developed a few years ago.31-33 Given a Hessianfrom anelectronicstructure calculation, a predictor step is takenonthe local quadratic surface. The Hessianis then recalculated and a fifthorder polynomial or a rational function is fittedto the energies, gradients and Hessians at the beginning and end points of this predictor step. The Bulrisch-Stoer algorithm21 is used to re-integrate the trajectory onthe fittedsurface toyield a correctorstep (see Figure 2). The process is repeated for each step. Since the Hessian at the end of the last step is used for the next predictor step, the electronic structure work is the same as for the secondorder Hessian-based method (z.e. one Hessian calculation per step). As shown in Figure 3, theerror inthe conservation of energy for the Hessian-based predictor corrector method isthreeordersofmagnitudelower thanfor the second order Hessian-based method, permitting a ten foldincrease in thestep size withoutloss of accuracyinthe energy conservation. This means an order of magnitude increase in the efficiency of theAIMDcalculation,sincethe number of electronic structure calculations for a given trajectory is reducedby a factor of ten.
Algorithms forgeometry optimization useupdating formulas to maintain and improve an estimated Hessian during an optimization.34,35 This approach can be applied to our Hessian-based predictor-corrector method for integrating trajectories. We have found that Bofill’s formula36 can be used toupdate theHessian for 5-10steps beforeit needs to be recalculated. As shown in Figure 4, this speeds up the trajectory integration by a factor of 3 or more for systems containing 4to 6 heavy atoms. Withupdating,the step size needs to beonly slightly smaller to maintain thesameenergy conservation as without updating. We have used the
Logarithm of mass-weighted step size(amu1/2bohr) Figure 3. Comparison of the error in the conservation of energy versus step size for trajectories integrated with the second order Hessian-based method (squares) and the Hessian-based predictor
corrector method with a fifth order polynomial (circles) or a rational function (triangles) for the corrector step (slopes of the least squares fits in parenthesis).
Number ofHessian니pdates
Figure 4. Relative cpu times as a function of the number of updates for Hessian-based Born-Oppenheimer trajectory calculations.
Hessian-based predictor-corrector method (with and without updating) in studies of H2CO T H2 + CO, F + C2H4 T C2H3F, C2H2O2 (glyoxal) T H2 + 2 CO & H2CO + CO, C2N4H2 (s-tetrazine) T N2 + 2 HCN and HXCOT HX+ CO.31。37-46
Collins has developed anovel method for growingpotential energy surfaces for dynamics by using trajectories to determine where additional electronic structure calculations are needed.47-53 An initial approximation to the potential energy surface is constructed with a modest number of
Figure 2. Details of the Hessian-based predictor-corrector algorithm for integrating classical trajectories.
energy, gradient and Hessian calculations alongthe reaction path. This localinformation is linked with distanceweighted interpolants to yield a global surface. As more trajectories are run, some explore regions of the surface farther away fromtheexisting data points.Additionalelectronicstructure calculations areperformed in these regions to improve the accuracy of the interpolated surface. Theprocess continues until the desired dynamical properties become stable with respect to improvements inthe surface. This approach has been usedby Collins and co-workers to study a numberof systems, including OH + H2, N+ H3+, BH++ H2 and triazine dissociation.54-60
Extended Lagrangian methods. Converging wavefunc tions for every time step in a trajectory calculation can be costly. Since relatively small time stepsare used, the change in thewavefunction may be small enough so thatit can be treated by suitable equations ofmotion. In 1985 Car and Parrinello20 outlined such an approach for ab initio molec ular dynamics (forreviews, see Ref. 61-63). Use ofthetime dependent Schrodinger equation was impractical, since it would necessitate very small time steps. Instead, they constructed an extendedLagrangian to obtainclassical-like equationsof motionfor thewavefunction:
L = 1/2Tr[ VTMV] + ^ xj |dQ./혜 2d r- E(R0)
- £*(
j
S ©dr-如 ⑴where R, Vand M are the nuclear positions, velocities and masses,卩 is the fictitious electronic mass, r are the elec tronic coordinates and A# are Lagrangian multipliers to ensure that theorbitals remain orthonormal. Thecoefficients of the molecular orbitals,奴 are expandedin a plane wave basis.64,65Thissimplifies many of theintegrals and facilitates applications to condensed matter. However, a very large number of plane waves is needed and the types of density functionals that can be used easily is limited (e.g. hybrid functionalsare expensive sincetheHartree-Fock exchangeis difficult to calculate). Furthermore, pseudopotentials must be used to replace core electrons, since these cannot be describe well by reasonable sized plane wave basis sets.
Even withthese limitations,theCar-Parrinelloapproachand its variants have seen extensive usage in the physics community.66
Molecular electronic structure calculations in chemistry are usually carried out with atom centered basis functions (e.g. gaussians) rather than plane waves.67-70 Since atom centeredbasis functions areautomatically positionedwhere the density is the greatest, far fewer functions are needed than plane waves. Fast integral packages are available for gaussian basis functions and hybrid density functionals are handled readily. Becausethedensitymatrix becomes sparse for large molecules, Hartree-Fock and density functional calculations can be made to scale linearly with molecular size.71-73 These features, coupled with the extensive experi ence that the chemistry community has with levels of theory and basis sets, lead us to develop the atom-centered
density matrix propagation (ADMP) method for molecular dynamics.74-76
The equations for propagation of the density matrix are simplest in an orthonormal basis. In many ways, this is similar to density matrix search methods for calculating electronic energies.77 In the ADMP approach, the extended Lagrangianfor thesystem is
L = 1/2Tr[ VTMV] + 1/2 Tr[(卩1/4W^1/4 )2 ]
-E(R,P) - Tr[A(PP- P)] (2) where P, W and 卩arethe densitymatrix, the densitymatrix velocity and the fictitious mass matrix for the electronic degrees of freedom. Constraints on the total number of electrons and the idempotency are imposed using the Lagrangian multiplier matrix A. The energy is calculated using the McWeeny purificationof the density,78 P =3 P2 - 2 P3.TheEuler-Lagrangeequationsof motion are
M d2R/dt2=- dE/dR|P；
Rd2P/dt2 =- [dE/dP|R+AP + PA - A] (3) These can be integrated usingthevelocity Verlet algorithm,64,79 P.+1 = P.+ Wi At - S1/2[dE(R,Pi)/dP|R + AP.+PiA. - A.]
x 广 At2/2
Wi+1/2 =Wi - r1/2 [dE(R,Pi)/dP|R+AB+ PA - Ai] x r-1/2At/2 = [Pi+1-PJ/At
Wi+1 = W+1/2 - r1/2 [dE(Ri+1,Pi+1)/dP|R+Ai+1Pi+1
+ Pi+1A 汁1 - Ai+1]冋项2At/2 (4) A simple iterative scheme is used to determine the Lagrangian multipliers so that Pi+1 and Wi+1 satisfy the idempotencyconstraints.74,75
Pi+1 —Pi+1 +广[Pi TPi+ (I-Pi)T(I-Pi)]『2 T
=R1/2 [ P ,+1-P,+1]R1/2
Wi+1 —Wi+1 +rT2 [P,+1TP,+1 + (I-P,+1)T(I-P,+1)]rT/2 T
=R1/2 [W,+1-W,+1]R1/2 (5) where W,+1 = P!+1W!+1(I-P,+1) +P!+1W!+1(I-P!+1). In calcu lating dE/dR|P we need to take into account that P is not converged andthat U, the transformation between thenon- orthogonal atomic orbital basis and the orthonormal basis, depends on R. This leads to a somewhat more complicated expression than used for gradients of converged SCF energies.
dE/dR|P=Tr[U-t dh'/dR|P U-1 P + U-tdG'(P )/HR|p U-1 P] -Tr[F dU/dR U-1 P + P U-tdU/dR F] + dFNVdR
- ~ ~ ~
=Tr[dh/dR\pP' +dG'(P')/dR|pP']
-Tr[F' U-1 dU/dR P'+ P'dU/dRU-t F'] + d PWdR (6)
Figure 5. Ratios of estimated timings for Born-Oppenheimer versus ADMP trajectory calculations on linear hydrocarbons, CnH2n+2, computed at HF/6-31G(d) (top) and B3LYP/6-31G(d) (bottom) with the Hessian-based predictor-corrector method (diamonds), Hessian-based predictor-corrector with updating (squares), gradient based velocity Verlet (triangles) and ADMP (diamonds).
where the primed quantities are integrals in the atomic orbital basis and U‘ U = S'. An important factor in the viability of thisapproachis that we havebeenable to obtain thederivativeofthe transformationmatrix inclosed formfor Choleskyorthonormalization.74
(dU/dR U-1)v=(U-‘ dS'/dR U-1)vfor卩 < v,
=1/2 (U-tdS'/dR U-1)vfor卩，=V,
=0 for卩，> v. (7)
Unlike earlier approaches topropagatingHartree-Fock and generalized valence bond wavefunctions.80-83 the ADMP method shows excellent energy conservation without thermostats and does not requireperiodicre-convergence of theelectronicstructure.
To estimate the relative timing ofthe BO and ADMP methodsfor molecular dynamics, we considered a series of linear hydrocarbons(seeFigure 5). OneFock matrix and one gradient evaluation per time step are neededin the ADMP approach. This is used as the reference for all the other methods.BO method with velocityVerlet usesapproximately the same time step as ADMP but needs an average of 10 Fockmatrix evaluationstoconverge thewavefunction. The Hessian-based trajectory integration methods can employ much larger time steps andstill maintain the same level of
energy conservation orbetter. When updating is used, the cost of calculatingthe Hessianis spread out over anumber of stepsthereby reducingtheaveragecostper step. As seen in Figure 5, this approach is most efficient for small moleculesandforcasesthat require moreaccuratedynamics.
The ADMP approachwins for larger systems and shows its advantageevenearlier for hybrid DFT methods.76
The ADMP method has some of the specific advantages and greaterflexibilitywhen compared to the Car-Parrinello approach. All electrons can be treated explicitly andpseudo
potentials are not required. Anydensityfunctional,including hybrid functionals, can be employed. Smaller fictitious masses can be used and goodadiabaticity can be maintained without thermostats.74-76 For ionic systems, vibrational frequencies calculated by the plane-wave Car-Parrinello method show a disturbing dependence on the fictitious electronicmass;84 however,the ADMPmethod isfreefrom this problem.76 The ADMP trajectories compare very well with those computed by BO methods.74 Specifically, for CH2O T H2 +CO and C2H2O2 T H2 + 2 CO, the ADMP trajectories give product translational, rotational and vibrationalenergy distributions that areveryclosetotheBO results.76 The ADMP is being extended to QM/MM treat ments for biologicalsystems, and hasbeen used to studythe solvationofexcessprotons inwater clusters andhydroxyl
stretch red shifts in chloride water clusters.76 Summary
Recent advances in computerhardware and software are making the applications of ab initio molecular dynamic increasingly more practical. Born-Oppenheimer methods offerthe advantage of propagating moleculeson welldefined potential energy surfaces. Extended Lagrangian methods yield very similar dynamics atareduced cost. The coming yearswill bring a rapidincreaseinthe number and types of systems that arestudied withthese approaches.
Acknowledgements. Ourcontributionstothe exploration of potential energy surfaceswere supported by the National Science Foundation, Wayne State University and Gaussian, Inc. I wouldlike tothank Professor Kwang S. Kim for the opportunity to participate in the 10th Korea Japan Joint Symposium on Theoretical and Computational Chemistry, and for organizingsuchasplendid meeting.
References
1. Allen, M. P; Tildesley, D. J. Computer Simulation of Liquids'； Oxford University Press: Oxford, England, 1987.
2. Haile, J. M. Molecular Dynamics Simulation: Elementary Methods； Wiley: New York, 1992.
3. Thompson, D. L. Trajectory Simulations of Molecular Collisions:
Classical treatment, in Encyclopedia of Computational Chemistry'； Schleyer, P. v. R., Allinger, N. L., Kollman, P. A., Clark, T., Schaefer III, H. F., Gasteiger, J., Schreiner, P. R., Eds.; Wiley:
Chichester, 1998; Vol. 5, pp 3056-3073.
4. Advances in Classical Trajectory Methods； Hase, W. L., Ed.; JAI
Press: Greenwich, 1992.
5. Bunker, D. L. Meth. Comput. Phys. 1971, 10, 287.
6. Raff, L. M.; Thompson, D. L. The Classical Trajectory Approach to Reactive Scattering, in Theory of Chemical Reaction Dynamics'； Baer, M., Ed.; CRC Press: Boca Raton, 1985.
7. Rapaport, D. C. The Art of Molecular Dynamics Simulation;
Cambridge University Press: Cambridge; New York, 1995.
8. Frenkel, D.; Smit, B. Understanding Molecular Simulation: from Algorithms to Applications, 2nd ed.; Academic Press: San Diego, 2002.
9. McCammon, J. A.; Harvey, S. C. Dynamics of Proteins and Nucleic Acids; Cambridge University Press: Cambridge Cambrid
geshire, New York, 1987.
10. Gunsteren, W. F. v.; Weiner, P. K.; Wilkinson, A. J.; Alliant Computer Systems Corporation. Computer Simulation of Bio- molecular Systems: Theoretical and Experimental Applications';
ESCOM: Leiden, 1989.
11. Kollman, P. Chem. Rev. 1993, 93,2395-2417.
12. Beveridge, D. L.; Dicapua, F. M. Annual Review of Biophysics and Biophysical Chemistry 1989, 18, 431-492.
13. Beveridge, D. L. Molecular Dynamics: DNA, in Encyclopedia of Computational Chemistry; Schleyer, P. v. R., Allinger, N. L., Kollman, P. A., Clark, T., Schaefer III, H. F., Gasteiger, J., Schreiner, P. R., Eds.; Wiley: Chichester, 1998; Vol. 3, pp 1620
1628.
14. Auffinger, P.; Weshof, E. Molecular Dynamics: Simulation of
15.
16.
17.
18.
Nucleic Acids, in Encyclopedia of Computational Chemistry;
Schleyer, P. v. R., Allinger, N. L., Kollman, P. A., Clark, T., Schaefer III, H. F., Gasteiger, J., Schreiner, P. R., Eds.; Wiley:
Chichester, 1998; Vbl. 3, pp 1628-1639.
Berendsen, H. J. C.; Tieleman, D. P. Molecular Dynamics: Studies of Lipid Bilayers, in Encyclopedia of Computational Chemistry;
Schleyer, P. v. R., Allinger, N. L., Kollman, P. A., Clark, T., Schaefer III, H. F., Gasteiger, J., Schreiner, P. R., Eds.; Wiley:
Chichester, 1998; Vbl. 3, pp 1639-1650.
Pedersen, L.; Darden, T. Molecular Dynamics: Techniques and Applications to Proteins, in Encyclopedia of Computational Chemistry; Schleyer, P. v. R., Allinger, N. L., Kollman, P. A., Clark, T., Schaefer III, H. F., Gasteiger, J., Schreiner, P. R., Eds.;
Wiley: Chichester, 1998; Vd. 3, pp 1650-1659.
Schatz, G. C. Rev. Mod. Phys. 1989, 61, 669-688.
Schatz, G. C. The Analytical Representation of Potential Energy Surfaces for Chemical Reactions, in Advances in Molecular Electronic Structure Theory; Dunning, T. H., Ed.; JAI Press:
London, 1990.
19. Bolton, K.; Hase, W. L.; Peslherbe, G. H. Direct Dynamics of Reactive Systems, in Modern Methods for Multidimensional Dynamics Computation in Chemistry; Thompson, D. L., Ed.;
World Scientific: Singapore, 1998; pp 143-189.
20. Car, R.; Parrinello, M. Phys. Rev. Lett. 1985, 55, 2471-2474.
21. Press, W. H. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed.; Cambridge University Press:
Cambridge, England; New York, NY, 1992.
22. Helgaker, T.; Uggerud, E.; Jensen, H. J. A. Chem. Phys. Lett.
1990,173,145-150.
23. Uggerud, E.; Helgaker, T. J. Am. Chem. Soc. 1992, 114, 4265- 4268.
24. Braten, S. M.; Helgaker, T.; Uggerud, E. Org. Mass Spectrom.
1993, 28,1262-1269.
25. Bueker, H. H.; Uggerud, E. J. Phys. Chem. 1995, 99, 5945-5949.
26. Bueker, H. H.; Helgaker, T.; Ruud, K.; Uggerud, E. J. Phys. Chem.
1996, 100,15388-15392.
27. Oiestad, A. M. L.; Uggerud, E. Int. J. Mass Spectrom. 1997, 167, 117-126.
28. Oiestad, E. L.; Uggerud, E. Int. J. Mass Spectrom. 1997, 165, 39- 47.
29. Ruud, K.; Helgaker, T.; Uggerud, E. Theochem-J. Mol. Struct.
1997, 393, 59-71.
30. Uggerud, E. Mass Spectrom. Rev. 1999, 18, 285-308.
31. Chen, W.; Hase, W. L.; Schlegel, H. B. Chem. Phys. Lett. 1994, 228, 436-442.
32. Millam, J. M.; Bakken, V.; Chen, W.; Hase, W. L.; Schlegel, H. B.
J. Chem. Phys. 1999, 111, 3800-3805.
33. Bakken, V.; Millam, J. M.; Schlegel, H. B. J. Chem. Phys. 1999, 111, 8773-8777.
34. Schlegel, H. B. Geometry Optimization, in Encyclopedia of Computational Chemistry; Schleyer, P. v. R., Allinger, N. L., Kollman, P. A., Clark, T., Schaefer III, H. F., Gasteiger, J., Schreiner, P. R., Eds.; Wiley: Chichester, 1998; Vol. 2, pp 1136
1142.
35. Schlegel, H. B. Geometry Optimization on Potential Energy Surfaces, in Modern Electronic Structure Theory; Yarkony, D. R., Ed.; World Scientific Publishing: Singapore, 1995; pp 459-500.
36. Bofill, J. M. J. Comput. Chem. 1994, 15, 1-11.
37. Vreven, T.; Bernardi, F.; Garavelli, M.; Olivucci, M.; Robb, M. A.;
Schlegel, H. B. J. Am. Chem. Soc. 1997, 119, 12687-12688.
38. Bolton, K.; Hase, W. L.; Schlegel, H. B.; Song, K. Chem. Phys.
Lett. 1998, 288, 621-627.
39. Bolton, K.; Schlegel, H. B.; Hase, W. L.; Song, K. Y. Phys. Chem.
Chem. Phys. 1999, 1, 999-1011.
40. Sanchez-Galvez, A.; Hunt, P.; Robb, M. A.; Olivucci, M.; Vreven, T.; Schlegel, H. B. J. Am. Chem. Soc. 2000, 122,2911-2924.
41. Li, X. S.; Millam, J. M.; Schlegel, H. B. J. Chem. Phys. 2000, 113, 10062-10067.
42. Bakken, V.; Danovich, D.; Shaik, S.; Schlegel, H. B. J. Am. Chem.
Soc. 2001, 123, 130-134.
43. Li, X. S.; Millam, J. M.; Schlegel, H. B. J. Chem. Phys. 2001, 114, 8897-8904.
44. Li, X. S.; Millam, J. M.; Schlegel, H. B. J. Chem. Phys. 2001, 115, 6907-6912.
45. Li, X. S.; Anand, S.; Millam, J. M.; Schlegel, H. B. Phys. Chem.
Chem. Phys. 2002, 4, 2554-2559.
46. Anand, S.; Schlegel, H. B. J. Phys. Chem. A 2002, 106, 11623
11629.
47. Collins, M. A. Theor. Chem. Acc. 2002, 108, 313-324.
48. Ischtwan, J.; Collins, M. A. J. Chem. Phys. 1994, 100, 8080-8088.
49. Jordan, M. J. T.; Thompson, K. C.; Collins, M. A. J. Chem. Phys.
1995, 103, 9669-9675.
50. Thompson, K. C.; Collins, M. A. J. Chem. Soc.-Faraday Trans.
1997, 93, 871-878.
51. Thompson, K. C.; Jordan, M. J. T.; Collins, M. A. J. Chem. Phys.
1998, 108, 8302-8316.
52. Thompson, K. C.; Jordan, M. J. T.; Collins, M. A. J. Chem. Phys.
1998, 108, 564-578.
53. Bettens, R. P. A.; Collins, M. A. J. Chem. Phys. 1999, 111, 816
826.
54. Jordan, M. J. T.; Thompson, K. C.; Collins, M. A. J. Chem. Phys.
1995, 102, 5647-5657.
55. Jordan, M. J. T.; Collins, M. A. J. Chem. Phys. 1996, 104, 4600
4610.
56. Bettens, R. P. A.; Collins, M. A. J. Chem. Phys. 1998, 109, 9728
9736.
57. Bettens, R. P. A.; Hansen, T. A.; Collins, M. A. J. Chem. Phys.
1999, 111, 6322-6332.
58. Collins, M. A.; Zhang, D. H. J. Chem. Phys. 1999, 111, 9924- 9931.
59. Fuller, R. O.; Bettens, R. P. A.; Collins, M. A. J. Chem. Phys.
2001, 114, 10711-10716.
60. Song, K. Y.; Collins, M. A. Chem. Phys. Lett. 2001, 335, 481-488.
61. Remler, D. K.; Madden, P. A. Mol. Phys. 1990, 70, 921-966.
62. Payne, M. C.; Teter, M. P.; Allan, D. C.; Arias, T. A.;
Joannopoulos, J. D. Rev. Mod. Phys. 1992, 64, 1045-1097.
63. Marx, D.; Hutter, J. Ab Initio Molecular Dynamics: Theory and Implementation, in Modern Methods and Algorithms of Quantum Chemistry; Grotendorst, J., Ed.; John von Neumann Institute for Computing: Julich, 2000; Vol. 1, pp 301-449.
64. Tuckerman, M. E.; Parrinello, M. J. Chem. Phys. 1994, 101,1302
1315.
65. Tuckerman, M. E.; Parrinello, M. J. Chem. Phys. 1994, 101,1316
1329.
66. Rothlisberger, U. Computational Chemistry 2001, 6, 33-68.
67. Hehre, W. J.; Radom, L.; Schleyer, P. v. R.; Pople, J. A. Ab Initio Molecular Orbital Theory Wiley: New York, 1986.
68. Foresman, J. B.; Frisch, A. Exploring Chemistry with Electronic Structure Methods, 2nd ed.; Gaussian Inc.: Pittsburgh, PA, 1996.
69. Jensen, F. Introduction to Computational Chemistry； Wiley:
Chichester, New York, 1999.
70. Cramer, C. J. Essentials of Computational Chemistry： Theories and Models; J. Wiley: West Sussex, England, New York, 2002.
71. Goedecker, S. Rev. Mod. Phys. 1999, 71, 1085-1123.
72. Scuseria, G. E. J. Phys. Chem. A 1999, 103,4782-4790.
73. Li, X. P; Nunes, W.; Vanderbilt, D. Phys. Rev. B 1993, 47, 10891- 10894.
74. Schlegel, H. B.; Millam, J. M.; Iyengar, S. S.; Voth, G. A.;
Daniels, A. D.; Scuseria, G. E.; Frisch, M. J. J. Chem. Phys. 2001,
114, 9758-9763.
75. Iyengar, S. S.; Schlegel, H. B.; Millam, J. M.; Voth, G. A.;
Scuseria, G. E.; Frisch, M. J. J. Chem. Phys. 2001, 115, 10291- 10302.
76. Schlegel, H. B.; Iyengar, S. S.; Li, X.; Millam, J. M.; Voth, G. A.;
Scuseria, G. E.; Frisch, M. J. J. Chem. Phys. 2002, 117, 8694
8704.
77. Millam, J. M.; Scuseria, G. E. J. Chem. Phys. 1997, 106, 5569- 5577.
78. McWeeny, R. Rev. Mod. Phys. 1960, 32, 335-339.
79. Swope, W. C.; Andersen, H. C.; Berens, P. H.; Wilson, K. R. J.
Chem. Phys. 1982, 76, 637-649.
80. Hartke, B.; Carter, E. A. J. Chem. Phys. 1992, 97, 6569-6578.
81. Hartke, B.; Carter, E. A. Chem. Phys. Lett. 1992, 189, 358-362.
82. Gibson, D. A.; Carter, E. A. J. Phys. Chem. 1993, 97, 13429
13434.
83. Gibson, D. A.; Ionova, I. V; Carter, E. A. Chem. Phys. Lett. 1995, 240, 261-267.
84. Tangney, P.; Scandolo, S. J. Chem. Phys. 2002, 116, 14-24.
